Gibbs's idea: the ensemble
A single system, followed in time, is hard to reason about. Gibbs's move was to replace time-averaging with an ensemble average: imagine a vast collection of identical copies of the system, one in each accessible microstate, all present 'at once' in our imagination. Averages over this frozen collection are far easier to compute, and — for an equilibrium system — they equal the long-time averages we actually measure.
An ensemble is defined by what it holds fixed and what it lets fluctuate. Three choices dominate physics, matching three physical situations: a truly isolated box, a box in a heat bath, and a box that exchanges both heat and particles with a reservoir.
The microcanonical ensemble
The microcanonical ensemble describes an isolated system with fixed energy E, volume V and particle number N. By the fundamental postulate, every one of its \Omega(E,V,N) microstates carries equal probability 1/\Omega. This is the ensemble in which Boltzmann's S=k_B\ln\Omega lives directly. It is conceptually cleanest — but computationally the hardest, because counting states at exactly energy E is fiddly.
In practice one counts states with energy up to E, call it \Phi(E), and the density of states g(E)=d\Phi/dE gives the number in a thin shell. For systems of many particles \Omega(E) climbs so steeply — like E^{3N/2} for an ideal gas — that whether you count the shell or everything below it makes no difference to \ln\Omega to leading order in N.
Temperature, pressure and chemical potential from entropy
Here is where counting becomes physics. Put two isolated systems in thermal contact so they can share energy but nothing else. The combined multiplicity is \Omega_1(E_1)\,\Omega_2(E-E_1); equilibrium sits at the energy split that maximizes it, i.e. maximizes S_1+S_2. Setting the derivative to zero gives \partial S_1/\partial E_1 = \partial S_2/\partial E_2. Whatever this shared quantity is, it is what equalizes at thermal equilibrium — it is temperature.
The three doors out of the microcanonical entropy. Differentiate S(E,V,N) with respect to each variable to get temperature, pressure and the chemical potential — the statistical definitions that match the Vol I thermodynamic ones.
These three relations define temperature, pressure and chemical potential microscopically. Notice the surprise in the first: temperature is inversely the slope of entropy versus energy. A system whose entropy rises steeply as you add energy (many new states open up) is cold — it eagerly absorbs energy. One whose \Omega barely grows is hot. Together these formulas reproduce the fundamental thermodynamic relation dE = T\,dS - P\,dV + \mu\,dN.
Worked count: the two-state paramagnet
To feel the machinery, count the states of N non-interacting spins, each either up or down, with n of them up. The number of microstates with a given n is the binomial coefficient — pure combinatorics:
The multiplicity of a spin macrostate. It peaks sharply at n=N/2 (equal up and down), which is why an unmagnetized state is overwhelmingly the most probable.
Using Stirling's approximation \ln N! \approx N\ln N - N, the entropy S=k_B\ln\Omega becomes a smooth function of the fraction x=n/N, peaking at x=1/2. Give the spins an energy (each up-spin costs \varepsilon in a field) and \partial S/\partial E immediately yields the temperature and the magnetization. This same binomial count, with different labels, describes defects in a crystal, mixing of two gases, and polymer conformations. Counting is universal.